At \(T_{\rm eff}\sim 300\)  K, the minimum liquid water content to explain most sub-Neptune masses and radii is \(>\sim 50\)  wt% even if there is no Fe-metal core [1]}. This is more H\(_2\) O than can possibly be produced by hydrogen-magma reactions [2]}, and instead implies a contribution of planet building blocks from the temperature range beyond the water ice snowline. This is a zone where (in the Solar System), abundant refractory carbon is found. Specifically, the carbon content of primitive chondrite meteorites (CI and CM type) is 2-6 wt% [3]}. Although we do not fully understand the origin of this refractory carbon, proposed mechanisms for forming this refractory carbon would also apply to exoplanetary systems [4]}. Therefore we assume a planet bulk composition of \({\rm (2-6\ wt\%)\times (1 - x)}\)  carbon, where \({\rm x}\)  is the H\(_2\) O mass fraction, and the remainder of the planet's building blocks are assumed to have a C content similar to that of primitive chondrites. This is a conservative lower limit on bulk C content for a thin-H\(_2\) -atmosphere sub-Neptune, for the following two reasons. (i) It considers only refractory C, not C ices (e.g., CO\(_2\)  ice) which could be important in the case of whole-planet migration. (ii) Some primitive bodies in the Solar System appear to be more C-rich than the most primitive chondrite meteorites; for example, the surface of the dwarf planet Ceres may contain 20 wt% C [5]}. These large bulk C contents map to substantial envelope C contents (Equations REF -REF ). As such, the \(4\times 10^{-4}\)  bar partial pressure of CO2, while not the absolute lower limit, is a cosmochemically and geologically reasonable lower bound of the CO2 partial pressure on an ocean planet.
